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1 In the format provided by the authors and unedited. DOI: 1.138/NPHOTON Supplementary Material: Scattering compensation by focus scanning holographic aberration probing (F-SHARP) Ioannis N. Papadopoulos 1, Jean-Sébastien Jouhanneau 2, James F. A. Poulet 2 Judkewitz 1 * and Benjamin 1: Bioimaging and Neurophotonics Lab, NeuroCure Cluster of Excellence, Charité Berlin, Humboldt University, Charitéplatz 1, 1117 Berlin, Germany 2: Max Delbrück Center for Molecular Medicine, Robert-Rössle-Str. 1, 1392 Berlin, Germany *corresponding author (benjamin.judkewitz@charite.de) Detailed description of experimental setup The output of a Ti:Sapphire laser (Mai Tai Deepsee, Spectra-Physics, USA) passes through a dispersion compensation unit and then through the combination of a λ/2 waveplate (HWP 1) and a polarizing beamsplitter (PBS 1) which acts as a variable attenuator. The laser beam is expanded 4-fold by the telescope comprising of lenses L 1 (achromat doublet f=75 mm, Thorlabs, USA) and L 2 (achromat doublet f=3 mm, Thorlabs, USA). The combination of the λ/2 waveplate (HWP 2) with the polarizing beamsplitter (PBS 2) controls the intensity ratio between the weak (aberrated) and the strong (corrected) beam used in F-SHARP. The strong beam is reflected by PBS 2 and then reflected off of the spatial light modulator (Pluto Phase Only SLM NIRII, Holoeye, Germany). The 5:5 nonpolarizing beamsplitter in front of the SLM (NPBS 2) acts as a circulator directing the beam off of the SLM towards the x-scanning galvo. Due to the distance between SLM and x-galvo (< 8 cm), any correction patterns are digitally propagated from the SLM plane to the galvo plane. The x-scanning galvo mirror (Cambridge Technologies, USA) is imaged onto the y-galvo scanning mirror (Cambridge Technologies, USA) by a pair of scan lenses (broadband scan lens, effective focal length=11 mm, Thorlabs, USA). The weak beam is reflected by a phase-stepping piezo mirror (Physik Instrumente, Germany), which is positioned onto a translation stage for path length matching. The reflection off of the phase stepper is directed via the non-polarizing Beamsplitter (NPBS 1) towards the tip-tilt piezo-scanner. The piezo-scanner (Piezo Tip/Tilt Mirror, Physik Instrumente, Germany) is responsible for the secondary scanning of the weak beam against the strong corrected beam during the E-field PSF estimation with F-SHARP. The two beams (weak and strong) are combined again through the polarizing beamsplitter (PBS 3). Since the two beams have orthogonal polarizations, they are then made to interfere by placing a polarizer after PBS 3. The pair of the scan lens (achromat doublet, f=75 mm, Thorlabs, USA) and the tube lens (achromat doublet, f=25 mm, Thorlabs, USA) image both the x and y galvo plane of the strong beam together with the tip-tilt piezo NATURE PHOTONICS 1

2 scanner plane of the weak beam onto the back aperture of the microscope objective. The objective lens (Nikon, 16x, Water Immersion, NA=.8 or Nikon 25x, Water Immersion, NA=1.1) focuses the excitation onto the focal plane. The generated fluorescent signal is captured by the same microscope objective and then is reflected by the primary dichroic mirror (longpass dichroic mirror, cutoff 678nm, Semrock, USA). The combination of lenses L 6 (achromat doublet, f=6 mm, Thorlabs, USA) and L 7 L 8 (aspheric lens, f=16 mm, Thorlabs, USA) images the back aperture of the objective onto the two photomultiplier tubes (Hamamatsu, Japan). A secondary dichroic mirror (longpass dichroic mirror, cutoff-wavelength 562 nm, Semrock USA) splits the fluorescent signal into two channels (red and green). M1 M2 L1 L2 beam expander NPBS 1 PBS 2 HWP 2 λ/2 waveplate PBS 1 beam dump M3 HWP 1 λ/2 waveplate phase-stepping piezo mirror NPBS 2 x-galvo scanner Ti:Sapphire Laser with Dispersion compensation power control tip-tilt piezo-scanner SLM M4 L3 L4 PBS 3 polarizer scan lens y-galvo scanner PMT 1 PMT 2 L7 tube lens primary dichroic L8 secondary dichroic L6 objective lens focal plane Figure S1. Detailed schematic of experimental setup. PBS - Polarizing beamsplitter, HWP - Half waveplate, M - mirror, L - lens, NPBS - nonpolarizing beamsplitter, SLM - spatial light modulator, PMT - photomultiplier tube. NATURE PHOTONICS 2

3 Mathematical analysis of F-SHARP In this supplement, we examine in detail the mathematical principle of F-SHARP and show that after each correction step the corrected E-field PSF at the focal plane will be equal to the 3 rd power of the previous correction step. We therefore prove that F-SHARP can turn any enveloped speckle pattern into a sharp focus spot within a finite number of correction steps. Assuming a uniform fluorescent sample, the signal captured onto a photodetector generated by the nonlinear interaction of a scanning and a stationary beam (in the case of two-photon (2P) absorption and within the memory effect range 1-3 ) will be proportional to I(x) x) + E stat ( x ) 4 d x. (1) The algebraic expansion formula for a + b 4, with a and b being complex-valued, is, a + b 4 = a 4 + b 4 +a 2 b *2 + a *2 b a 2 b 2 +2 a 2 ab * + 2 a 2 a * b + 2 b 2 ba * + 2 b 2 b * a. (2) Setting the ratio between the two beams such that E stat 2 / 2 <.1, we can discard all the factors in Equation 1 that contain the weak beam, E stat, in powers equal to and larger than 2. Therefore, Equation 1 reads I(x) x) 4 d x + 2 E ( * scan +2 x) 2 ( x x)e * ( x )d x. stat x x) 2 x)e stat ( x )d x (3) In order to isolate the second term in Equation 2 we employ a phase stepping scheme similar to plane wave interferometry where the phase between the object and the reference beam, in this case between the scanning and the stationary beam, is changed in a number of steps (minimum of 2, usually 4) around the unit circle. Setting the phase difference between the two beams at Δφ i =,π / 2,π,3π / 2 we get, I i (x) x) 4 d x + 2 E ( scan +2 ( x x) 2 * x)e stat ( x )e iδφ i d x x x) 2 x)e * stat ( x )e iδφ i d x. (4) We may then compute, E f (x) = (I I π ) + i (I π /2 I 3π /2 ) with the 4 measurements, to isolate the second term, E f (x) x) 2 * ( x x)e ( x )d x. (5) stat This is the optical field we recover in the first correction step of F-SHARP. Next we examine E f (x) in the Fourier domain. To do so, we perform the first variable substitution of x x in Equation 5: E f ( x) ( x + x) 2 * ( x + x)e ( x )d x. (6) stat NATURE PHOTONICS 3

4 The flipped reconstructed complex field after the phase stepping operation, E f (-x), is now a crosscorrelation between the field product (x) 2 * (x) and E stat (x). Using the cross-correlation Fourier theorem, the Fourier transform of Equation 6 produces { } = F (x) 2 * F E f ( x) and the complex conjugate of Equation 7 will be { } * = F (x) 2 * F E f ( x) { (x)} * F E stat (x) { (x)} F E stat (x) { }, (7) { } *. (8) plane wave inhomogeneous medium aberrated E-field PSF E PSF (x) Fourier transform back aperture focusing lens focusing plane input. F{E PSF (x)} aberrated E-field PSF E PSF (x) Figure S2 Left, The propagation of a plane wave incident on the back aperture of a focusing lens through an inhomogeneous medium at the focal plane results in the generation of a scattered E-field, E PSF (x). Right, The propagation of any input on the back Fourier plane of a focusing lens to the focal plane is equivalent to the forward Fourier transform, F { }, of the multiplication of the input with the inverse Fourier transform of E PSF. We have that E at focus (x) = F 1 { input F { E PSF (x)}}. The above analysis is based on scalar diffraction theory. Although high numerical aperture (NA) microscope objectives are known to induce polarization rotation 4, such an effect is negligible for linearly polarized excitation light and the numerical apertures considered here (< 1% 2P signal contributed by new polarizations for NA < 1.1 in water). The goal of F-SHARP is to optically correct the scattering and aberrations induced by an inhomogeneous medium, using optical phase conjugation. In other words, we would like to be able to form a sharp focus through the inhomogeneous medium. Now, we will demonstrate how the multiple step measurement and optical phase conjugation (OPC) correction of E f (x) will yield an increasingly sharp focus. Let's assume that we have measured E f (x). The propagation of a plane wave through a focusing lens and an inhomogeneous medium that lies between the lens and the focal plane will result in a complex field distribution at the focal plane, which we call E PSF (x). When any input field, incident on the back aperture of the focusing lens, propagates through the inhomogeneous medium towards the focal plane, the field distribution at the focal plane is equivalent to the inverse Fourier NATURE PHOTONICS 4

5 transform of the product between the input field at the back aperture and the Fourier transform of E PSF (x), F { E PSF (x)}, as shown in Figure S2. Moreover, the stationary beam appearing in the equations above, is generated by a plane wave at the back aperture of the objective lens, which means that it is equal to the scattered E-field PSF: E stat (x) = E PSF (x). If we now use a spatial light modulator to introduce the complex conjugate of the Fourier transform of the flipped measurement E f (x), as in Equation 8, into the Fourier plane of the imaging system, the field distribution at the focal plane will be equal to E at focus plane (x) = F 1 F (x) 2 * { (x)} F { E PSF (x)} *!###### "###### $ F { E (x)} PSF!#" # $. (9) input scattering Here we have substituted E stat (x) with E PSF (x). In an optical propagation system such as the one shown in Figure S2, the time reversal property of light propagation implies that, F { E PSF (x)} * F { E PSF (x)} = 1. (1) Therefore, from Equation 9 we have that, the use of the complex conjugate of the Fourier transform of the flipped measurement as the input to the system, will turn the corrected beam at the focal plane equal to, E at focus plane (x) = (x) 2 * (x) = (x) 3 e iφ scan (x), (11) After performing optical phase conjugation, the corrected beam amplitude will be proportional to the cube of the scanned beam amplitude used during the previous measurement. During the first measurement, the scanning field is also equal to the scattered E-field PSF: (x) = E PSF (x). Therefore, after the corrections are applied during the first correction step, the corrected scanning beam will be equal to E corr (x) = E PSF (x) 3 e iφ PSF (x). (12) Now, we will repeat this process. The corrected field here, E corr (x), will now be scanned against the stationary scattered PSF: E PSF (x). We may thus insert it into Equation 1 to redefine the function (x). We may follow exactly the same analysis to find that the newly corrected beam at the focal plane will be E (x) = E corr, 2 nd iter ( (x) 3 PSF ) 3 e iφ PSF (x). (13) Now generalizing, the field at the focal plane after the j th correction step will be equal to NATURE PHOTONICS 5

6 E corr, j th iter (x) = E PSF (x) 3j e iφ PSF (x). (14) We can see that by correcting the scanning beam after each measurement step, E corr will converge towards the third power of the corrected PSF of the previous step. Applying this nonlinear factor on any aberration field will help any dominant mode to prevail against the weaker side lobes and therefore we can generate a diffraction-limited PSF at the focusing plane. It is clear from this analysis that the number of correction steps needed to converge towards a diffraction-limited corrected PSF will strongly depend on the original shape of the scattered PSF. The presence of a dominant mode will allow us to converge faster, compared to a case when multiple modes have comparable intensities. Moreover, the above analysis, although presented for the case of two-photon absorption and therefore two-photon fluorescence imaging, is not limited to this nonlinear interaction. It will hold that for any nonlinear effect of order n the fluorescent signal will be proportional to ( ) n d I(x) x) + E stat ( x ) 2 x. (15) Setting the power ratio between scanning and stationary beams again such that only the powers of 2n and 2n 1 of the algebraic expansion will contribute to the final signal it follows that the corrected E-field PSF after each correction step will be proportional to E (x) = E (x) (2n 1) j corr, j th iter PSF e iφ PSF (x) (16) and thus, the amplitude of E corr will be taken to the power of 2n-1 with each correction step. For example in case of 3P fluorescence ( n = 3 ) the convergence rate will be proportional to the 5 th power. The mathematical derivation was done in a one-dimensional case for simplicity and can be easily extended to a two dimensions without any difficulties. However, in the above analysis we have not considered any volume effects. In 2P imaging the second order nonlinear dependence of the generated signal with the excitation light intensity makes the off-focus contributions insignificant. This is the reason for the increased sectioning capabilities of 2P microscopy. In the case of increased scattering, the generated speckle pattern becomes more uniform along z. In this case, out-of-focus contributions cannot be neglected any more and F-SHARP measurements will contain contributions from all the different planes where the envelope of the speckle pattern is strong. The reconstructed correction pattern will be superimposed on top of background contributions that constitute noise. With the sample and the scattered E-field PSF becoming more extended along z, the efficiency of the scattering compensation will decrease 5. In this scenario higher order nonlinearities, eg. 3P fluorescence, are expected to enhance the correction process. NATURE PHOTONICS 6

7 Comparison between F-SHARP and pixel optimization iterative techniques State of the art scattering compensation techniques operate by iterating through the pixels of a wavefront shaper (SLM) to estimate the phase pattern that will correct for optical scattering. Different methods have been developed for iterating through the modes of an SLM, like sequential pixel scanning 6, Hadamard mode scanning 7, genetic algorithms 5,8 and frequency modulation 9. A variation of the frequency modulation technique splits the excitation beam into a modulated and an unmodulated part The advantage of this approach is that the unmodulated part acts as a reference beam inside the medium. In this case any residual ballistic light can act as a guide star. However, splitting the beam into equal parts between the reference and modulated beam on the wavefront shaper results in a 2-fold decrease in the amplitude of the reference beam focus, causing a 16-fold decrease in the number of 2P generated photons from the ideal focus location (4x drop in shot-noise limited signal-to-noise ratio, SNR). F-SHARP does not split the reference beam on the spatial light modulator and thus avoids this reduction in focus intensity. The way that iterative techniques scan through the available wavefront pixel modes affects their behavior. When the pixels are scanned sequentially 6, the interference between the stationary pixels and the modulated one is very weak. Thus this type of technique will be significantly affected by shot noise. On the contrary, when the modulation of the pixel phase value occurs in the temporal frequency domain 9,1, the modulation is stronger but results in the generation of beat frequencies. These beat frequencies have to be discarded, also reducing the fraction of photons that contribute useful signal. F-SHARP does not require rejection of beat frequencies. a b normalized diffraction efficiency of pixelated grating Measurement time (in sec) pixel-based scanning 2.5kHz pixel-based scanning 1kHz F-SHARP galvo 1kHz F-SHARP resonant 8kHz x in μm (in focal plane) Number of modes Figure S3 a. Diffraction efficiency of a pixelated SLM along the focus plane. The calculation was performed for λ=92 nm, objective NA=.8, and 32 pixel SLM along one dimension. The intensity diffraction efficiency of pixelated devices drops to ~4% for the highest order. b. Measurement time needed for MEMS vs. galvo scanning scattering compensation. Galvo scanned F-SHARP scattering compensation scales with the square root of the number of modes and therefore scaling favorably compared to pixel optimization iterative algorithms. Attempting to reconstruct a continuous wavefront with a pixelated device (even within the Nyquist limit) will suffer from decreased efficiency at steeper wavefront slopes (which is the same NATURE PHOTONICS 7

8 principle why a binary phase grating diffracts with a 4% efficiency at the 1 st order). The angular diffraction efficiency of the pixelated wavefront shaper will be modulated by a sinc 2 function, therefore exhibiting reduced detection and modulation efficiency towards the higher order modes (higher angles, see Figure S3a). F-SHARP directly measures the E-field PSF at the focus plane without modulating with a pixelated device and then compensates for the scattering by reconstructing the continuous phase wavefront at the Fourier plane with a very high pixel count SLM. This allows F- SHARP both to avoid pixelisation effects during its PSF measurement step and correct higher order modes with increased efficiency compared to pixel-based iterative optimization techniques. The measurement time for the aforementioned techniques depends linearly on the number of measured modes. Decoupling the measurement of the scattered field from the speed of the wavefront shaper allows F-SHARP to accelerate the measurement time especially as the number of corrected modes increases. F-SHARP measures the E-field PSF over a 2 dimensional space and does that with a fixed horizontal linescan rate. The acquisition rate of a single image, containing N number of modes, scales with the number of lines of the final image. Thus, the measurement time per mode for F-SHARP is proportional to the square root of the number of modes, therefore scaling favorably with larger number of modes (Figure S3b). References 1. Freund, I., Rosenbluh, M. & Feng, S. Memory effects in propagation of optical waves through disordered media. Phys. Rev. Lett. 61, (1988). 2. Feng, S., Kane, C., Lee, P. & Stone, A. Correlations and fluctuations of coherent wave transmission through Disordered Media. Phys. Rev. Lett. 61, (1988). 3. Judkewitz, B., Horstmeyer, R., Vellekoop, I.M., Papadopoulos, I.N. & Yang, C. Translation correlations in anisotropically scattering media. Nature Phys. 11, (215). 4. Kang, H., Jia, B. & Gu, M. Polarization characterization in the focal volume of high numerical aperture objectives. Opt. Express 18, (21). 5. Katz, O., Small, E., Guan, Y. & Silberberg, Y. Noninvasive nonlinear focusing and imaging through strongly scattering turbid layers. Optica 1, (214). 6. Vellekoop, I.M. & Mosk, A.P. Focusing coherent light through opaque strongly scattering media. Opt. Lett. 32, (27). 7. Popoff, S. M. et al. Exploiting the Time-Reversal Operator for Adaptive Optics, Selective Focusing, and Scattering Pattern Analysis. Phys. Rev. Lett. 17, (211). 8. Conkey, D.B., Brown, A.N., Caravaca-Aguirre, A.M. & Piestun, R. Genetic algorithm optimization for focusing through turbid media in noisy environments. Opt. Express 2, (212). 9. Bridges, W.B. et al. Coherent optical adaptive techniques. Appl. Opt. 13, (1974). 1. Tang, J., Germain, R.N. & Cui, M. Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique. Proc. Natl. Acad. Sci 19, (212). 11. Kong, L. & Cui, M. In vivo fluorescence microscopy via iterative multi-photon adaptive compensation technique. Opt. Express 22, (214). 12. Park, J.-H., Sun, W. & Cui, M. High-resolution in vivo imaging of mouse brain through the intact skull. Proc. Natl. Acad. Sci. U.S.A. 112, (215). NATURE PHOTONICS 8

9 DOI: 1.138/NPHOTON a b c d π/2 Figure S4 Alignment of SLM to the back aperture of the microscope objective. a, Phase pattern projected onto the SLM (background phase:, letter n: π). b, Correct mapping of the back aperture to the SLM plane through an affine transformation leads to a cancelling of the applied phase pattern and therefore a flat phase. c, Reconstructed E-field PSF with F-SHARP by imaging a uniform fluorescent sample. d, Fourier transform of reconstructed E-field PSF provides a map of the back aperture of the objective. NATURE PHOTONICS 9

10 a c b d Figure S5 Estimation of measured modes using F-SHARP. a, Fourier transform of reconstructed E-field PSF. The number of modes contained in the phase pattern projected onto the SLM is measured as the ratio of the diameter of the correction phase pattern over the mean modal size. The mean modal size is defined as the Full Width to Half Maximum (FWHM) of the autocorrelation of the correction pattern. b, The autocorrelation is computed. c,d, The FWHM along the two axis is calculated. The ratio between the original diameter and the mean modal diameter yields a number of recovered and corrected modes equal to NATURE PHOTONICS 1